Related papers: On eigenvalues of double branched covers
We show for an alternating knot the minimal boundary slope of an essential spanning surface is given by the signature plus twice the minimum degree of the Jones polynomial and the maximal boundary slope of an essential spanning surface is…
We investigate the negative definite spin fillings of branched double covers of alternating knots. We derive some obstructions for the existence of such fillings and find a characterization of special alternating knots based on them.
We classify all knot diagrams of genus two and three, and give applications to positive, alternating and homogeneous knots, including a classification of achiral genus 2 alternating knots, slice or achiral 2-almost positive knots, a proof…
We show that if $p/q$-surgery on a nontrivial knot $K$ yields the branched double cover of an alternating knot or link, then $|p/q|\leq 4g(K)+3$. This generalises a bound for lens space surgeries first established by Rasmussen. We also show…
The non-orientable 4-genus of a knot K in the three sphere is defined to be the minimum first Betti number of a non-orientable surface F in the four-ball so that K bounds F. We will survey the tools used to compute the non-orientable…
We will strengthen the known upper and lower bounds on the delta-crossing number of knots in therms of the triple-crossing number. The latter bound turns out to be strong enough to obtain (unknown values of) triple-crossing numbers for a…
We give an upper bound for the dealternating number of a closed 3-braid. As applications, we determine the dealternating numbers, the alternation numbers and the Turaev genera of some closed positive 3-braids. We also show that there exist…
Roberts proved that a family of alternating, arborescent, prime knots each have at least $2^{2n-1}$ distinct minimal genus Seifert surfaces, where $n$ is the genus of the knot in question. We give a subfamily of these knots that have…
In this paper, we develop a lower bound for the double slice genus of a knot using Casson-Gordon invariants. As an application, we show that the double slice genus can be arbitrarily larger than twice the slice genus. As an analogue to the…
In formulating a non-orientable analogue of the Milnor Conjecture on the $4$-genus of torus knots, Batson developed an elegant construction that produces a smooth non-orientable spanning surface in $B^4$ for a given torus knot in $S^3$.…
We investigate the question of when distinct branched surfaces in the complement of a 2-bridge knot support essential surfaces with identical boundary slopes. We determine all instances in which this occurs and identify an infinite family…
We prove that for knots, the evaluation of the Jones polynomial at the sixth root of unity, as well as the evaluation of the $Q$-polynomial at the reciprocal of the golden ratio, are uniquely determined by the oriented homeomorphism type of…
The $\Delta$-unknotting number for a knot is defined as the minimum number of $\Delta$-moves needed to deform the knot into the trivial knot. We determine the $\Delta$-unknotting numbers for two-bridge knots of type $C(2\beta_1, 2\beta_2,…
The non-orientable 4-genus of a knot in the 3-sphere is defined as the smallest first Betti number of any non-orientable surface smoothly and properly embedded in the 4-ball, with boundary the given knot. We compute the non-orientable…
Given a knot in the 3-sphere, the non-orientable 4-genus or 4-dimensional crosscap number of a knot is the minimal first Betti number of non-orientable surfaces, smoothly and properly embedded in the 4-ball, with boundary the knot. In this…
We give sharp two-sided linear bounds of the crosscap number (non-orientable genus) of alternating links in terms of their Jones polynomial. Our estimates are often exact and we use them to calculate the crosscap numbers for several…
We present a practical algorithm to determine the minimal genus of non-orientable spanning surfaces for 2-bridge knots, called the crosscap numbers. We will exhibit a table of crosscap numbers of 2-bridge knots up to 12crossings (all 362 of…
We show that, for an alternating knot, the ratio of the diameter of the set of boundary slopes to the crossing number can be arbitrarily large.
The non-orientable 4-genus of a knot $K$ in $S^{3}$, denoted $\gamma_4(K)$, measures the minimum genus of a non-orientable surface in $B^{4}$ bounded by $K$. We compute bounds for the non-orientable 4-genus of knots $T_{5, q}$ and $T_{6,…
We present a lower bound on the stable $4$-genus of a knot based on Casson-Gordon $\tau$-signatures. We compute the lower bound for an infinite family of knots, the twist knots, and show that a twist knot is torsion in the knot concordance…